Related papers: Some Motzkin-Straus type results for non-uniform h…
In 1965, Motzkin and Straus established a remarkable connection between the order of a maximum clique and the Lagrangian of a graph and provided a new proof of Tur\'an's theorem using the connection. The connection of Lagrangians and…
A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in [7]. This connection and its extensions were successfully employed in optimization to provide heuristics…
A remarkable connection between the order of a maximum clique and the Graph-Lagrangian of a graph was established by Motzkin and Straus in 1965. This connection and its extension were useful in both combinatorics and optimization. Since…
Motzkin and Straus established a close connection between the maximum clique problem and a solution (namely graph-Lagrangians) to the maximum value of a class of homogeneous quadratic multilinear functions over the standard simplex of the…
There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus in 1965. It is useful in practice if similar results hold for hypergraphs. However the obvious generalization of Motzkin…
There is a remarkable connection between the maximum clique number and the Lagrangian of a graph given by T. S. Motzkin and E.G. Straus in 1965. This connection and its extensions were successfully employed in optimization to provide…
Motzkin and Straus established a remarkable connection between the maximum clique and the Lagrangian of a graph in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum…
In 1965, Motzkin and Straus [5] provided a new proof of Turan's theorem based on a continuous characterization of the clique number of a graph using the Lagrangian of a graph. This new proof aroused interests in the study of Lagrangians of…
It is conjectured by Frankl and F\"uredi that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform hypergraphs with $m$…
The hypergraph jump problem and the study of Lagrangians of uniform hypergraphs are two classical areas of study in the extremal graph theory. In this paper, we refine the concept of jumps to strong jumps and consider the analogous problems…
An $r$-uniform graph $G$ is dense if and only if every proper subgraph $G'$ of $G$ satisfies $\lambda (G') < \lambda (G)$, where $\lambda (G)$ is the Lagrangian of a hypergraph $G$. In 1980's, Sidorenko showed that $\pi(F)$, the Tur\'an…
Unlike graphs, determining Tur\'{a}n densities of hypergraphs is known to be notoriously hard in general. The essential reason is that for many classical families of $r$-uniform hypergraphs $\mathcal{F}$, there are perhaps many…
The Lagrangian density of an $r$-uniform hypergraph $F$ is $r!$ multiplying the supremum of the Lagrangians of all $F$-free $r$-uniform hypergraphs. For an $r$-graph $H$ with $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge…
The uniform Tur\'an density $\pi_{u}(F)$ of a $3$-uniform hypergraph (or $3$-graph) $F$ is the supremum of all $d$ such that there exist infinitely many $F$-free $3$-graphs $H$ in which every induced subhypergraph on a linearly sized vertex…
In the 1980s, Erd\H{o}s and S\'os initiated the study of Tur\'an hypergraph problems with a uniformity condition on the distribution of edges, i.e., determining density thresholds for the existence of a hypergraph H in a host hypergraph…
Generalized Tur\'an problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem, maximizing the number of cliques of a fixed order in a graph with fixed number of vertices and…
We study Tur\'an-type extremal problems for distance graphs, motivated by work of Csikv\'ari, Bollob\'as, Tyomkyn, and Uzzell. We determine the maximum number of vertex pairs at distance three in an $n$-vertex graph with no triangle formed…
The study of uniform Tur\'an densities was initiated in the 1980s by Erd\H{o}s and S\'os. Given a $3$-graph $F$, the uniform Tur\'an density of $F$, $\pi_{\therefore}(F)$, is defined as the infimum $d\in[0,1]$ such that every $3$-graph $H$…
In 1965, T. S. Motzkin and E. G. Straus established an elegant connection between the clique number of a graph and the global maxima of a quadratic program defined on the standard simplex. Over the years, this seminal finding has inspired a…
Generalized Tur\'an problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem is maximizing the number of cliques of size $t$ in a graph of a fixed order that does not contain…